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1. Chapter 7
Dimensional Analysis,
Similitude,
and Modeling

2. John Smeaton(1724-1792)first used scale models for systematic experimentation. William Froude (1810-1871) first proposed laws for estimating ship hull drag from model tests. Aimee Vaschy, Lord Rayleigh, D. Riabouchinsky, E. Buckingham all made significant contributions to dimensional analysis and similitude. Jean B. J. Fourier (1768-1830) first formulated a theory of dimensional analysis. Osborne Reynolds (1842-1912)first used dimensionless parameters to analyze experimental results. Moritz Weber (1871-1951)assigned the name Reynolds number and Froude number. HISTORICAL CONTEXTIntroduction

3. Introduction
zThere remain a large number of problems that rely on experimentally obtained data for their solution. zThe solutions to many problems is achieved through the use of a combination of analysis and experimental data. zAn obvious goal of any experiment is to make the results as widely applicable as possible. zConcept of similitude model prototype V7.1 Real and model fliesIt is necessary to establish the relationship between the laboratory model and the actual system, from which how to best conduct experiments and employ their results can be realized.

4. 7.1 Dimensional Analysis
zConsider Newtonian fluid through a long smooth-walled, horizontal, circular pipe.
Determine pressure dropppx∂ Δ= ∂l
pressure drop per unit length
(),,,pfDVρμΔ=lHow can you do for the last two cases?

5. Dimensional Analysis
zConsider two non-dimensional combinations of variables zThe results of the experiment could then be represented by a single universal curve. zThe curve would be valid for any combination of smooth walled pipe, and incompressible Newtonian fluid. 2DpVDV ρφρμ ⎛⎞Δ=⎜⎟ ⎝⎠ l

6. Dimensional Analysis
zTo obtain this curve we could choose a pipe of convenient size and fluid that is easy to work with. zThe basis for this simplification lies in the consideration of the dimensions of the variable involved. zThis type of analysis is called dimensional analysis which is based on Buckingham pi theorem. () ()() ()()() () 3000224214210002LFLDpFLTVFLTLTFLTLTLVDFLTFLT ρρμ −− −− − Δ== == l& && 34221pFLFLTDLFLTVLT ρμ − − − Δ== == = l& && &

7. 7.2 Buckingham pi theorem
zHow many dimensionless products are required to replace the original list of variables?
“If an equation involving kvariables is dimensionally homogeneous, it can be reduced to a relationship among k –rindependent dimensionless products, where ris the minimum number of reference dimensions required to describe the variables.” The dimensionless products are frequently referred to as “pi terms,”and the theorem is called the Buckingham pi theorem. () () 123123,, ,, kkrufuuu φ− = Π=ΠΠΠLLThe required number of pi terms is fewer than the number of original variables by r, where ris determined by the minimum number of reference dimensions required to describe the original list of variables. MLT, FLT

8. 7.3 Determination of pi terms
zmethod of repeating variables
1: List all the variablesthat are involved in the problem.
Geometry of the system (such as pipe diameter)
Fluid properties (ρ, μ)
External effects (driving pressure, V)
It is important that all variables be independent.
2: Express each of the variables in terms of basic
dimensions.
MLT, FLT22342FmaMLTFMLTMLFLTρ − − −− == = ∴==

9. Determination of pi terms
3: Determine the required member of pi terms. Buckingham pi theorem: kvariablesrreference dimensions (M, L, T, or θ) k –r independent dimensionless groups4: Select a number of repeating variables, where the number required is equal to the number of reference dimensionsNotes: 1. Each repeating variable must be dimensionallyindependent of the others. 2. Do not choose the dependent variable (e.g., Δp) as oneof the repeating variables. ⇒

10. Determination of pi terms
5: Form a pi form by multiplying one of the nonrepeatingvariables by the product of the repeating variables, each raised to an exponent that will make the combination dimensionless. 6: Repeat Step 5 for each of the remaining nonrepeatingvariables. 7: Check all the resulting pi terms to make sure they are dimensionless. : nonrepeatingvariable: repeating variables 123,,,abciuuuuiu123,,abcuuu

11. Determination of pi terms
8: Express the final form as a relationship among the pi terms, and think about what it meansThe actual functional relationship among the pi terms must be determined by experiments. ()123,,krφ−Π=ΠΠΠL

12. Determination of pi terms
zReconsider pipe pressure drop problem zk=5, basic dimensions: FLT r=3, pi terms: 5-3=2() 34221,,, ,,,, pfDVpFLDLFLTFLTVLT ρμρμ−−−− Δ= Δ===== ll&&&& ()()() 13142000121013402201abcbcapDVFLLLTFLTFLTcFaabcLbbcTcpDV ρρ −−− Π=Δ= +== −++−==− −+==− ΔΠ= ll&

13. ()()() ()() ()() ()() 22142000230001224212000214221012401 1201abcbcaDVFLTLLTFLTFLTcFaabcLbDVbcTcFLLpDFLTVFLTLTFLTFLTDVLLTFLTpVDorVDV μρμρρμρμρφφρρμ −−− − −− − −− Π= = +==− −++−==−→Π= −+==− ΔΠ=== Π=== ⎛⎞⎛⎞Δ∴=⎜⎟⎜⎟ ⎝⎠⎝⎠ ll& && && %
Determination of pi termsDetermination terms

14. Ex 7.1 Determine drag
hw() 21131,,,, 633fwhVMLTwLhLMLTMLVLTkr μρμρ − −− − − = = = = = = = −=−= & & & & & & DD ,,wVρ
repeating variable 1222322,, abcabcabcwVwVhhwVwwVwVhwVwwVwwVh ρρρμμρρμρφφρρμ Π== Π== Π== ⎛⎞⎛⎞ ==⎜⎟⎜⎟ ⎝⎠⎝⎠ % DDD V7.2 Flow past a flat plate

15. 7.4.1 Selection of variables
zIf extraneous variables are included, then too many pi terms appear in the final solution.
zIf important variables are omitted, then an incorrect result will be obtained.
zUsually, we wish to keep the problem as simple as possible, perhaps even if some accuracy is sacrificed.
zA suitable balance between simplicity and accuracy is a desirable goal.

16. Selection of variables
zFor most engineering problems, pertinent variables can be classified into three general groups. Geometry: such as length, diameter, etc. Material properties: External Effects: zSince we wish to keep the number of variables to a minimum, it is important that all variables are independent. zIf we have a problem, and we know that then qis not requiredand can be omitted. But it can be considered separately, if needed. ,ρμ ,Vg(),,,,,,0fqruvwρ=L()1,,,qfuvw=L

17. 7.4.2 Determination of Reference
Dimensions
zThe use of FLT or MLT as basic dimensions is the simplest.
zOccasionally, the number of reference dimensions needed to describe all variables is smaller than the number of basic dimensions. (e.g., Ex. 7.2)
Ex. 7.2

18. 7.4.3 Uniqueness of Pi Terms
zConsider pressure in a pipe
Select D, V, ρasrepeating variables
zIf instead choosing D, V, μas repeating variables
zTherefore there is not a unique set of pi termswhich arises from a dimensional analysis. 2pDVDV ρφρμ ⎛⎞Δ=⎜⎟ ⎝⎠ l21pDVDV ρφμμ ⎛⎞Δ=⎜⎟ ⎝⎠ l

19. Uniqueness of Pi Terms
zHowever, the required number of pi terms is fixed, and once a correct set is determined all other possible sets can be developed from this set by combinations of products of powers of the original set. () () () 1232231123122222Thus, where,arearbitraryexponentsThen, , ababpDpDVDVV φφφρρμμ Π=ΠΠ′Π=ΠΠ′Π=ΠΠ′Π=ΠΠ⎛⎞⎛⎞⎛⎞ΔΔ=⎜⎟⎜⎟⎜⎟ ⎝⎠⎝⎠⎝⎠ ll

20. Uniqueness of Pi Terms
zThere is no simple answer to the question:
Which form for the pi terms is best?
zUsually, the only guideline is to keep the pi terms as simple as possible.
zAlso, it may be that certain pi terms will be easier to work with in actually performing experiments.

21. 7.6 Common Dimensionless7.6 DimensionlessGroups in Fluid MechanicsGroups Mechanics

22. zFroude number, FrVinertiaforceFrgravitationalforceg== lIsssssFamdVdVaVdtds= == where sis measured along the streamline
Common Dimensionless
Groups in Fluid Mechanics

23. zIf Vsand sare expressed in dimensionless form **,ssVsVsV== lwhere and represent some characteristic velocity and length.Vl*2* * *2* * *2* * , sssssssIssGsIsGdVdVdVVaVVdtdsdsdVVFmaVmFmgdsdVFVVFgdsVFrg=== === = = llllFor a problem in which gravity(or weight) is not important, the Froude number would not appear as an important pi term.
Common Dimensionless
Groups in Fluid Mechanics
V7.4 Froudenumber

24. zReynolds number zEuler numberinertiaforceReviscousforceVρμ =lRe<<1: creeping flowLarge Re: the flow can be considered nonviscous22pressureforceEuinertiaforceppVVρρ Δ== Some form of the Euler number would normally be used in problems in which pressure or the pressure difference between two points is an important variable. For problem in which cavitationis of concern, 212rppV υρ − : cavitationnumber Common DimensionlessCommon DimensionlessGroups in Fluid MechanicsGroups MechanicsV7.3 Reynolds number

25. www.boattest.com/images-gallery/News/prop_cav.jpgwww.amhrc.edu.au/images/cavtunnel- propellor2.jpeg Cavitationis the formation of vapor bubbles of a flowing liquid in a region where the pressure of the liquid falls below its vapor pressure. For example, cavitationmay occur when the speed of the propeller tip is so high that the liquid pressure becomes lower than the vapor pressure212rppV υρ − : cavitationnumber

26. Common Dimensionless
Groups in Fluid Mechanics
zCauchy Numberand Mach Number222Ca,MaMaCaEVVVcEcEVE υυυυρρρρ ==== == When the Mach number is relatively small (less than 0.3), the inertial forces induced by the fluid motion are not sufficientlylarge to cause a significant change in the fluid density, and inthis case the compressibility of the fluid can be neglected.

27. Common Dimensionless
Groups in Fluid Mechanics
zStrouhalNumberStV ω =limportant in unsteady, oscillating flow problem in which the frequency of the oscillation is ω. It represent a measure of the ratio of inertia forces due to theunsteadiness of the flow (local acceleration, ) to the inertia forces due to change in velocity from point to point in the flow field (convective acceleration, ).
singing wires
V7.5 StrouhalnumbertV∂∂/ /VVx∂∂

28. Common Dimensionless
Groups in Fluid Mechanics
zWeber Number2WeVρσ =lIt is important when the surface tensionat the interface between two fluids is significant. Surface tension is aline force (F/L), whose effects become significant, even dominant, when the scale decreases to about < 100 μm. V7.6 Weber number

29. 7.7 Correlation of
Experimental data
zOne of the most important uses of dimensional analysis is as an aid in the efficient handling, interpretation, and correlation of experimental data.
zDimensional analysis provide only the dimensionless groupsdescribing the phenomenon, and not the specific relationship among the groups.
zTo determine this relationship, suitable experimental data must be obtained.

30. 7.7.1 Problems with One Pi Term
1CΠ=
where C is a constant
7.7.2 Problems with Two or More Pi Terms ()12φΠ=Π()123,φΠ=ΠΠEx. 7.3Flow with only one Pi termEx. 7.4Dimensionless correlation of experimental dataV7.7 Stokes flow

31. 7-8 Modeling and Similitude
zA model is a representation of a physical system that may be used to predict the behavior of the system in some desired respect. zThe physical system for which the predictions are to be made is called the prototype. zUsually a model is smaller than the prototype. Occasionally, if the prototype is very small , it may be advantageous to have a model that is larger than the prototype so that it can be more easily studied.V7.9 Environmental models

32. 7.8.1 Theory of models
zIt has been shown that
()123,,nφΠ=ΠΠΠL
If the equation describes the behavior of a particular prototype, a similar relationship can be written for a model of this prototype, ie.
()123,,mmmnmφΠ=ΠΠΠL
where the form of the function will be the same as long as the same phenomenon is involved in both the prototype and the model.

33. Theory of models
zTherefore, if the model is designed and operated under the following conditions,
2233, , mmnmnΠ=ΠΠ=ΠΠ=ΠM
model design condition or similarity requirement or modeling laws
then with the presumption that the form of is the same for model and prototype, it follows that
11mΠ=Π
-prediction equation

34. (),,,,fwhVμρ=D
pi theorem indicates2222, ,mmmmmmmmmmwVwwVhwVwwVh ρφρμρφρμ ⎛⎞ =⎜⎟ ⎝⎠ ⎛⎞ =⎜⎟ ⎝⎠ DD
model design conditions2222222, , thus, or, mmmmmmmmmmmmmmmmmmmmwVwwVwhhhwwwVVhwwVwVwVwV ρρμμμρμρρρρρ == == = ⎛⎞⎛⎞⎛⎞ =⎜⎟⎜⎟⎜⎟ ⎝⎠⎝⎠⎝⎠ DDDD Thus, to achieve similarity between model and prototype behavior, all the corresponding pi terms must be equated between model and prototype.
Theory of models

35. Theory of models
Similarity:
zGeometric similarity: (length scale)
A model and prototype are geometrically similar if and only if all body dimensions in all three coordinates have the same linear-scale rate. (including angles)
zKinematic similarity: (length scale and time scale,
ie. velocity scale)
The motions of two systems are kinematicallysimilar if homologous particles lie at homologous points at homologous times
zDynamic similarity
Model and prototype have the same length-scaleratio, time-scaleratio, and force-scale(mass-scale) ratio
Ex. 7.5V7.10 Flow past an ellipse

36. 7.8.2 Model Scales
zThe ratio of like quantities for the model and prototype naturally arises from the similarity requirements. 11221122lengthscalevelocityscaleetc. mmmmVVVV= = llll

37. 7.8.3 Practical Aspects of Using Models
zValidation of Model DesignIt is desirable to check the design experimentally whenever possible. May run tests with a series of models of different sizes. zDistorted ModelIf one or more of the similarity requirements are not met, e.g., , then it follows that the prediction equation, is not true, i.e., . Models for which one or more of the similarity requirements are not satisfied are called distorted models. 11mΠ=Π22mΠ≠Π11mΠ≠ΠV7.12 Distorted river model

38. Practical Aspects of Using Models
ze.g., open channel or free surface flowRe,VVFrg ρμ ==llFroude number similarity: mmmmmVVVVgg λ=→==l llllReynolds number similarity: mmmmmmmmmmVVVV ρμνρρμμμρν =→==llllll()32mmmmmvVvV λ⇒===lllllllThe common fluid used is water, therefore the above requirement will not be satisfied.

39. 7.9 Some Typical Model Studies
7.9.1 Flow Through Closed Conduits
zExample: flow through valves, fittings, metering devices.
zFor low Mach numbers (Ma < 0.3), any dependent Pi term (such as pressure drop) can be expressed as,
ilDependent pi term,,iVερφμ ⎛⎞ =⎜⎟ ⎝⎠ llllwhere ε: surface roughness; : a particular length dimension; , i=1,2, …: a series of length terms of the system,,imimmmmmmmVVερερμμ ===llllllllIf the pressure drop is the dependent variable then, 1222,mmmpppVVVρρρ ΔΔΔΠ== Ex. 7.6l

40. zFor large Reynolds numbers, inertial forces >>viscous forces, and in this case it may be possible to neglect viscous effects; i.e., it would not be necessary to maintain Reynolds number similarity between model and prototype. However, both model and prototype have to operate at large Reynolds number, and the dependent pi term ceases to be affected by changes in Re. (will be shown later) zFor flows cavitationphenomenon, then the vapor pressure becomes an important variable and an additional similarity requirement such as equality of the cavitationnumber is required, where is some reference pressure. pυ 212()/rppVυρ−rpFlow Through Closed ConduitsFlow Conduits

41. 7.9.2 Flow Around Immersed Bodies
zFlow around aircraft, automobiles, golf balls , and buildings.
Dependent pi term,,iVερφμ ⎛⎞ =⎜⎟ ⎝⎠ llllFrequently drag is of interest.D 2212,,iDVCV ερφρμ ⎛⎞ ==⎜⎟ ⎝⎠ lllll DThus•geometric similarity:,imimmm εε ==llllllmmmmVVρρμμ =ll•Reynolds number similarity: To reduce Vmwaterair1 (/~0.1),or (increase )mmp νννρρν →<> mmmmmmVVV μνρμρν →==llll>1V7.14 Model airplane test in water

42. Flow Around Immersed Bodies zFortunately, in many situations the flow characteristics are notstrongly influenced by Re over the operating range of interest. For high Re, inertial forces are dominant, and CDis essentially independent of Re(Fig. 7.7--CD for a sphere). 2122,4DCAVAd ρπ = = D

43. zNASA Ames 40 ×80 ft 345 mil/hr12 ×24 m 552 km/hrtest sectionFlow Around Immersed BodiesEx. 7.7V7.15 Large scale wind tunnel

44. zFor problem of high Mach number (Ma>0.3), compressibility effect grows significant. zIn high speed aerodynamics the prototype fluid is usually air, c= cm, ν= νm, and it is difficult to satisfy the above condition, for reasonable length scales. zThus, models involving high speed flows are often distorted with respect to Reynolds number similarity, but Mach number similarity is maintained. mmVVcc= Combined with Reynolds number similarity mmmcc νν →=llFlow Around Immersed Bodies

45. 3/8 scale wind tunnel test for automobiles : →240km/h wind speed to match Re(no compressibility effect concern) 1/8 scale wind tunnel test for trucks or buses : →700km/h wind speed to match Re(compressibility effects arise!) →match of Re unwanted. (Incomplete similarity) Then, how to solve this problem? Use Re-independence of drag coeff. above a certain Re. EX. 7.5 of Fox, et al. “Introduction to Fluid Mechanics,”7thEd., 2010. --Incomplete Similarity: Aerodynamic Drag on a BusIncomplete Similarity: Incomplete Automobile and Truck Tests

46. 7.9.3 Flow with a Free Surface
Weber numberDependent pi term2,,,,iVVVg ερρφμσ ⎛⎞ =⎜⎟⎜⎟ ⎝⎠ llllllSo thatFroude number: ,mmmmmmVVVggVgg λ==→==lllllReynolds number:() 3232mmmmmmVVρνρλμμν ⎛⎞=→==⎜⎟ ⎝⎠lllllWeber number:()2mmσρλσρ =l zFor large hydraulic structures, such as dam spillways, the Reynolds numbers are large, viscous forces are smallin comparison to the forces due to gravity and inertia. Therefore, Re similarity is not maintained and model designed on the basis of Froude number (why not Re?). Ex. 7.8(7.15) V7.19 Dam model

47. Model Tests(Fig. 7.2) (a) Measure the total drag coeff. (CD,T)mfrom model tests at corresponding Fr(b) Calculate analytically the friction drag coeff. (CD,F)m, (c) Find the wave drag coeff. (CD,W)m=(CD,T)m-(CD,F)m. How to find the full-scale ship resistance from model test results?Follow the following procedure:(from Fox, et al. “Introduction to Fluid Mechanics”) Incomplete Similarity: Flow with a Free SurfaceIncomplete SurfaceV7.20 Testing of large yacht model

48. Prototype Predictions(Fig. 7.3) (a) Match (CD,W)p= (CD,W)mat corresponding Fr (Froude no. scaling), (b) Calculate analytically (CD,F)p, (c) Find (CD,T)p=(CD,W)p+ (CD,F)p. Note: Special treatment (adding studs) is needed for the ship model to stimulate turbulent boundary layer at proper position. Flow with a Free SurfaceFlow Surface

49. 7.10 Similitude Based on
Governing Differential Equations
zConsider 2-D equation 222222220uxyuuupuuutxyxxypugtxyyxy υρυμυυυυυρυρμ ∂∂ += ∂∂ ⎛⎞⎛⎞∂∂∂∂∂∂ ++=−++⎜⎟⎜⎟∂∂∂∂∂∂⎝⎠⎝⎠ ⎛⎞⎛⎞∂∂∂∂∂∂ ++=−+++⎜⎟⎜⎟∂∂∂∂∂∂⎝⎠⎝⎠ new dimensionless variables: 0*,*,* *,*,*:referencelength, :referencetimeupupVVpxytxyt υυττ === ===lll

50. zTherefore, 22222*** ** and*** *** uVuxVuxxxxuVuxVuxxxxx∂∂∂∂ == ∂∂∂∂ ∂∂∂∂∂⎛⎞==⎜⎟∂∂∂∂∂⎝⎠ lllThus{{ { 222022220**0** ******** ****** ****** **** GIPIcuxypVuVuupVuuutxyxxypVVpVugtxyyFFFF υρρμυτρυρυυμυρτ ∂∂ += ∂∂ ⎡⎤⎛⎞⎛⎞∂∂∂∂∂∂⎡⎤⎡⎤⎡⎤++=−++⎜⎟⎜⎟⎢⎥⎢⎥⎢⎥⎢⎥∂∂∂∂∂∂⎣⎦⎣⎦⎣⎦⎝⎠⎣⎦⎝⎠ ⎡⎤⎛⎞∂∂∂∂⎡⎤⎡⎤++=−++⎜⎟⎢⎥⎢⎥⎢⎥∂∂∂∂⎣⎦⎣⎦⎝⎠⎣⎦ llllll123{ 22222** ** VxyF υυ⎛⎞∂∂⎡⎤+⎜⎟⎢⎥∂∂⎣⎦⎝⎠linertia (local) forceinertia (convective) forcepressure forcegravity forceviscous force

51. 2202222202222******** ****** ******** ****** puuupuuuVtxyVxVxyppguVtxyVyVVxy μυτρρυυυμυυυτρρ ⎛⎞⎡⎤⎡⎤∂∂∂∂∂∂⎡⎤++=−++⎜⎟⎢⎥⎢⎥⎢⎥∂∂∂∂∂∂⎣⎦⎣⎦⎣⎦⎝⎠ ⎛⎞⎡⎤⎡⎤∂∂∂∂∂∂⎡⎤⎡⎤++=−+++⎜⎟⎢⎥⎢⎥⎢⎥⎢⎥∂∂∂∂∂∂⎣⎦⎣⎦⎣⎦⎣⎦⎝⎠ llllla form of StrouhalnumberEuler numberreciprocal ofFroude number
square
reciprocal ofReynolds number zIf two systems are governed by these equations, then the solutions (in terms of u*,υ*,p*, x*, y*, andt*) will be the same if the four parameters: (B.C.smust also be the same). 202,,,pVVVVg ρτρμ lllare equal for the two systems.